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Ptolemy and Copernicus

As far as I can determine, at least from his Little Commentary, Copernicus's big beef with Ptolemy seems to have been his use of the equant. Now, I have read the Almagest, many years ago, but I'm pretty sure that Ptolemy's system only required 12 circles, not the up to 240 claimed by some science writers.. Copernicus, at the end of his essay, says that his system requires 34 circles. So, my question is, does a Ptolemaic system without an equant require more circles than Copernicus's? If not, I can't really understand what Copernicus was trying to do.

"Occam" is the name of the alien race that will enslave us all eventually. And they've got razors for hands. I don't know if that's true but it seems like the simplest answer."

I can't speak to the accuracy of either system, other than to say that neither system was at all accurate by today's standards. Neither was the key issue some attempt to get working approximations with the least degree of complexity-- you get what you pay for, so if you want more accuracy, you will need more complexity. Copernicus may have achieved some improved efficiency by having the orbits centered on the correct object, but he gave back a lot of that accuracy by requiring that the orbits be circles, which of course is way off for several planets. I don't think the story of Copernicus vs. Ptolemy should be framed as an attempt to achieve greater accuracy, nor even an attempt to achieve greater simplicity. Instead, this is what I would say it was really all about, even if part of this is in a hindsight that Copernicus did not have access to:

1) By placing the Sun at the center, Copernicus made the first step toward explaining why orbits behave as they do-- in particular, the role of the mass of the object ruling those orbits. In his day, no one knew which had more mass, the Earth or the Sun, but the Sun certainly had a very significant role in the solar system, so it made some degree of sense to allow it to rule the orbits. Of course, later on, it was quite a breakthrough to look for what aspect of the Sun gave it that power.

2) By putting the Earth in motion, Copernicus allowed the Earth to be unified with the other planets. This allowed us to start looking for laws that were obeyed on Earth as a way to understand other planets as well, and even the entire cosmos. This was such a huge breakthrough its importance should never be overlooked.

3) Having the Earth in motion, yet not showing stellar parallax, required that the distant stars be so far away that they themselves were like Suns of their own. This was another vastly important unification of the cosmos, it allowed us to understand other stars by using the Sun as a basic example. It also allowed us to speculate on the possibility of, and then search for, planets around distant stars.

So that's what Copernicus was really doing, I just don't know how much of that he himself actually realized at the time. Certainly the issue of whether a proper coordinate system placed the Earth at the center, or the Sun, was a question that went completely out the window with general relativity. Unfortunately, that last issue tends to be the one many people focus on, though it is utterly irrelevant in comparison to the above three.

The problem for Ptolemy wasn’t epicycles, but was that he didn’t know the distances to the planets. But in place of that ignorance he provided six variables for correction for those who might find the correct distances. Copernicus’ system was more complex since he used 48 epicycles whereas Ptolemy used 40. Do we still use “epicycles” today, but they are called by different names, such as Fourier analysis? It is next to impossible to determine the exact revolutions of the planets due to their constant perturbations. We can estimate them, but they are never exact, since we have no math beyond a three-body problem to calculate them.

I don't think Ptolemy's lack of proper distance ratios would have been an issue for the outer planets in his geocentric model, but the light curves of Mercury and Venus would have been seriously at odds with his construction which kept them inside the Sun's sphere. It would appear that he did not pay attention to their brightness variations. For all we know he may not have even recognized that they shine by reflected sunlight.

Here is a nice paper on this topic. Only 6 epicycles and 6 deferents are needed, apparently, given today's knowledge.

Iphone

Good find, George. Clearly the equant-regulated eccentric circle was an excellent approximation of the Kepler ellipse at the relatively low eccentricities of the planets' orbits. While Copernicus, with his insistence on returning to the ideal of uniform circular motions or combinations thereof, made more sense mechanically in the paradigm which predated a gravitational dynamic theory, it appears that Ptolemy would have been significantly more accurate had he not messed up in some of his constructions.

To demonstrate the idea of making sense mechanically, we could make a mechanical model of the Copernican system with small potter's wheels spinning on bearings that are carried on the rims of larger wheels. The planets would be bright spots on the small wheels. In a thought exercise with no friction, it would run forever once set in motion. Ptolemy would require a rod rotating around the equant, with the planet sliding in and out radially on the rod while riding on a deferent track of some sort. Mechanically it would be a nightmare compared with the Copernican wheels.

Good find, George. Clearly the equant-regulated eccentric circle was an excellent approximation of the Kepler ellipse at the relatively low eccentricities of the planets' orbits.

Yes, that's interesting since we, or me at least, thought the big break-through was in Kepler's discovery of the elliptical orbits. This improved the accuracy, which was intended to match Tycho's accurate data of about 1 arcminute, though a few objects were about 1/2 arcminute in accuracy. The key is the eccentricity (an offset from Earth for the center of the radial vector).

While Copernicus, with his insistence on returning to the ideal of uniform circular motions or combinations thereof, made more sense mechanically in the paradigm which predated a gravitational dynamic theory, it appears that Ptolemy would have been significantly more accurate had he not messed up in some of his constructions.

Since Copernicus did use eccentric orbits by offsetting the center of the orbits from the center of the Sun, I would expect he may have been just as accurate. It seems also, though I haven't read all that I need of the paper, that, like Fitzpatrick's model, the apparent size of the Sun and other objects are a better fit with Cop's model.

Since Copernicus did use eccentric orbits by offsetting the center of the orbits from the center of the Sun, I would expect he may have been just as accurate.

Copernicus' main reason for opposing Ptolemy’s model was not because Ptolemy put the Earth in the center, but because he departed from the Greek practice of using perfect circles to depict the revolutions of the planets:

"In both De revolutionibus and the Commentariolus Copernicus attacks the Ptolemaic astronomy not because in it the sun moves rather than the earth, but because Ptolemy has not strictly adhered to the precept that all celestial motions must be explained only by uniform circular motions or combinations of such circular motions. Ptolemy had recognized that an accurate representation of planetary motion necessitated the abandoning of uniform circular motion, and he boldly introduced what was later called an “equant,” from which non-uniform motion along an arc would appear uniform. From the point of view of accuracy, this was a great step forward, indeed, the best representation of planetary motion before Kepler. But Copernicus considered the use of an equant to be a violation of fundamental principles and devoted his original astronomical research to devising a system of sun, planets, moon, and stars in which the planets and the moon glide with uniform motion along a circle or with some combination of such motions." (I. Bernard Cohen, Revolution in Science, 1994, p.112)

Copernicus' main reason for opposing Ptolemy’s model was not because Ptolemy put the Earth in the center, but because he departed from the Greek practice of using perfect circles to depict the revolutions of the planets:

"In both De revolutionibus and the Commentariolus Copernicus attacks the Ptolemaic astronomy not because in it the sun moves rather than the earth, but because Ptolemy has not strictly adhered to the precept that all celestial motions must be explained only by uniform circular motions or combinations of such circular motions. Ptolemy had recognized that an accurate representation of planetary motion necessitated the abandoning of uniform circular motion, and he boldly introduced what was later called an “equant,” from which non-uniform motion along an arc would appear uniform. From the point of view of accuracy, this was a great step forward, indeed, the best representation of planetary motion before Kepler. But Copernicus considered the use of an equant to be a violation of fundamental principles and devoted his original astronomical research to devising a system of sun, planets, moon, and stars in which the planets and the moon glide with uniform motion along a circle or with some combination of such motions." (I. Bernard Cohen, Revolution in Science, 1994, p.112)

[my bold] But they both used perfect circles, Ptolemy used constant angular motion for the radial vector but offsetting the Earth (equant) from the center of the orbit, which produced the non-uniform motion along the orbit, relative to the Earth. Cop favored uniform motion along the orbit and I assume he thought the equant was ad hoc and without physical reasoning. I am fairly sure each planet had to have its own equant, multiplying ad hocness.

It's interesting to me that Ptolmey's model required, IIRC, two major overhauls to produce usable tables during Cops younger years. These constant updates may have prompted interest in seeking an alternative model. The elegant and simple explanation for retrogrades had to be another big plus when he put his model to work. Having support from others, including one or more cardinals, probably helped as well, at least in moral support.

Just as a side note: I don't think that Ptolemy, or Hipparcos thought that celestial objects moved in perfect circles, but rather that perfect circles were easy (or at least possible) to compute, hence the epicycles on circles approach as a good approximation. Kepler's big advance wasn't that he guessed it was ellipses, it was his hard work developing the arithmetic enabling the calculation of positions along the ellipses.

Yes, that's interesting since we, or me at least, thought the big break-through was in Kepler's discovery of the elliptical orbits. This improved the accuracy, which was intended to match Tycho's accurate data of about 1 arcminute, though a few objects were about 1/2 arcminute in accuracy. The key is the eccentricity (an offset from Earth for the center of the radial vector).

Since Copernicus did use eccentric orbits by offsetting the center of the orbits from the center of the Sun, I would expect he may have been just as accurate. It seems also, though I haven't read all that I need of the paper, that, like Fitzpatrick's model, the apparent size of the Sun and other objects are a better fit with Cop's model.

The presentation on pp. 71-73 in the paper makes it clear to me that Ptolemy and Copernicus differ from Kepler by the same amount in the position angle calculation, but that Copernicus is off by twice as much in the distance from the Sun. At or near quadrature Copernicus's resultant is about as far outside Ptolemy's circle as that circle is outside the Kepler ellipse. I could see it clearly when drawing the Copernican model with a compass at e = 0.2. That difference between the two models would be immaterial for naked eye or low power telescopic views of the planets from the Sun's position, but when looking at Mars at quadrature with Earth near the Martian line of apsides there would be a significant difference in the angular position of Mars.

The Copernican resultant is approximately an ellipse with its major axis aligned with Kepler's minor axis, and its minor axis coinciding with Kepler's major axis.

Clear as mud? If necessary I can scan and upload some sketches. Sometimes a picture is worth many words.

Just as a side note: I don't think that Ptolemy, or Hipparcos thought that celestial objects moved in perfect circles, but rather that perfect circles were easy (or at least possible) to compute, hence the epicycles on circles approach as a good approximation. Kepler's big advance wasn't that he guessed it was ellipses, it was his hard work developing the arithmetic enabling the calculation of positions along the ellipses.

I'm unsure why perfect circles wouldn't be favored, if not required? Shootin' from the hip...perfection was of above; the simplest path around is a circle; crystaline spheres are spherical (you like that one?); ellipses were exotic geometric shapes occurring only if you cut a cone improperly (georgecentricity running amuck). Why would uniform motion be important if ellipses were tolerated; a crooked path should allow for variable speeds, perhaps?

[Added: Is it just me that hears "eccentric" and an ellipse comes to mind? I had to become clear what they meant by that term since it was used by the Greeks and yet Kepler took several years to install his ellipses.]

The presentation on pp. 71-73 in the paper makes it clear to me that Ptolemy and Copernicus differ from Kepler by the same amount in the position angle calculation, but that Copernicus is off by twice as much in the distance from the Sun. At or near quadrature Copernicus's resultant is about as far outside Ptolemy's circle as that circle is outside the Kepler ellipse. I could see it clearly when drawing the Copernican model with a compass at e = 0.2. That difference between the two models would be immaterial for naked eye or low power telescopic views of the planets from the Sun's position, but when looking at Mars at quadrature with Earth near the Martian line of apsides there would be a significant difference in the angular position of Mars.

A drawing would be helpful and if I had some free time I would like to try it on my own, but time is a problem for me these days.

I think I may be confused on this issue, or perhaps the Sun's size is a separate story. Perhaps I misunderstood the following found near the bottom of page 8 (after the diagram):

"Nevertheless, the Hippachian model is incorrect, since it predicts too large (by a factor
of 2) a variation in the radial distance of the sun from the earth (and, hence, the angular size of the
sun) during the course of a year (see Cha. 4). Ptolemy probabaly adopted the Hipparchian model
because his Aristotelian leanings prejudiced him in favor of uniform circular motion whenever this
was consistent with observations. (It should be noted that Ptolemy was not interested in explaining
the relatively small variations in the angular size of the sun during the year—presumably, because
this effect was difficult for him to accurately measure.)"

I'm unsure why perfect circles wouldn't be favored, if not required? Shootin' from the hip...perfection was of above; the simplest path around is a circle; crystaline spheres are spherical (you like that one?); ellipses were exotic geometric shapes occurring only if you cut a cone improperly (georgecentricity running amuck). Why would uniform motion be important if ellipses were tolerated; a crooked path should allow for variable speeds, perhaps?

I'm not sure when perfect crystalline spheres became the meme. but Ptolemy never mentioned it. He was merely trying to explain how to predict where things would be, and was pretty aware that the formulas weren't quite right, especially for Mars and Mercury.

I'm not sure when perfect crystalline spheres became the meme. but Ptolemy never mentioned it. He was merely trying to explain how to predict where things would be, and was pretty aware that the formulas weren't quite right, especially for Mars and Mercury.

Yes. [I just added to my post as you were making this post.] You may be right, but even his epicycles were circles. His equant will produce a tight fit for any ellipse, but the non-uniform motion hurts that modeling except for near-circular orbits, which the Sun and Earth have with one another. If a geometric model without initial physical restraints was used, why not have tried ellipses? That may be asking too much of Ptolmey but he was likely a genius.

Good thing Ptolemy et al. liked circular epicycles. A retrograde elliptical epicycle with 2:1 axial ratio does a stunning job of approximation Keplerian motion for even pretty substantial eccentricity (and relaxing the axial ratio and ratio of periods gives a pattern that works well for stellar orbits within galaxies).

Here is my sketch. The solid arc is a Kepler ellipse of eccentricity e = 0.2, about that of Mercury. The short-dashed arc is what Ptolemy's circle should have been, here transformed to a heliocentric model and sharing geometric center CPK with the ellipse. The long-dashed arc is Copernicus's circular deferent with an epicycle riding on it and with its geometric center at CC. The radius vectors from the Sun's position to the ellipse are shown for 1/8 and 1/4 of the orbital period from perihelion. The corresponding radius vectors for Ptolemy and Copernicus are plotted with dashed lines.The three dots for each time show where the planet is in the respective model, with Copernicus being the resultant of the deferent and epicycle vectors. As mentioned in the paper, the epicycle revolves in the same direction at twice the rate. To reduce visual clutter, I did not bother to plot the Copernicus resultant arc, but the two points I did plot are clearly about twice as far from the ellipse as are the corresponding Ptolemy points. The Kepler angles were calculated from the author's equation in the paper.

ETA: Use the thumbnail at the top. I don't know how the bottom one got there, and I cannot find a way to delete it.

Clearly the equant-regulated eccentric circle was an excellent approximation of the Kepler ellipse at the relatively low eccentricities of the planets' orbits.

“The equant got Ptolemy into a lot of trouble as far as many of his successors were concerned. It wasn’t that his model didn’t predict the
angular positions satisfactorily. Rather, the equant forced the epicycle to move non-uniformly around the deferent circle, and that was somehow seen as a deviation from the pure principle of uniform circular motion. Ptolemy himself was apologetic about it, but he used it because it generated the motion that was observed in the heavens. Altogether his system was admirably simple considering the apparent
complexity and variety of the retrograde loops” ('The Book that Nobody Read', O.Gingerich p. 53).

Here is my sketch. The solid arc is a Kepler ellipse of eccentricity e = 0.2, about that of Mercury. The short-dashed arc is what Ptolemy's circle should have been, here transformed to a heliocentric model and sharing geometric center CPK with the ellipse. The long-dashed arc is Copernicus's circular deferent with an epicycle riding on it and with its geometric center at CC. The radius vectors from the Sun's position to the ellipse are shown for 1/8 and 1/4 of the orbital period from perihelion. The corresponding radius vectors for Ptolemy and Copernicus are plotted with dashed lines.The three dots for each time show where the planet is in the respective model, with Copernicus being the resultant of the deferent and epicycle vectors. As mentioned in the paper, the epicycle revolves in the same direction at twice the rate. To reduce visual clutter, I did not bother to plot the Copernicus resultant arc, but the two points I did plot are clearly about twice as far from the ellipse as are the corresponding Ptolemy points. The Kepler angles were calculated from the author's equation in the paper.

ETA: Use the thumbnail at the top. I don't know how the bottom one got there, and I cannot find a way to delete it.

Thanks Hornblower for the graph. I want to study your points closer but thought I would try a few things graphically since it is fairly easy to do and I am pressed for time today.

Your drawing does indeed match the eccentricity you state. The blue dashed line is an ellipse with e = 0.2.

My sketches cause me to question the author of the paper where he says that Ptolemy and Copernicus are in agreement on the true anomaly when calculated to 2nd order in equations 4.31 and 4.35 respectively. He did not say how small the eccentricity had to be for the approximation to be "good enough", and he did not show how he did the expansions. I am guessing a Taylor series, but it has been 50 years since I studied the technique. If I am not mistaken, that 2nd order expression is truncated from an infinite series. I found an online calculator for Kepler. http://www.jgiesen.de/kepler/kepler1.html It shows the expansion to 5th order along with a Newtonian calculation to 8 decimal places. Taking it to 5th order reduced my true anomaly at 1/4 of the orbital period by about half a degree, with a vanishingly small difference from Newton. Since I don't know how to do the expansion I am unable to test it for Ptolemy and Copernicus, but I could see differences between them in my graphical plots. Those appear to be exact constructions, and the two pairs of points appear to be exactly parallel with the respective deferent radius vectors and thus not aligned with the Sun. Of course I cannot draw to that level of precision but I could see what was happening geometrically.

In making some more sketches, I find myself between a rock and a hard place trying to improve the fit between Copernicus and Kepler. In the sketch I posted, the two constructions have the same eccentricity. Copernicus is too slow in angular velocity around the Sun around perihelion, and the resultant orbital radius is too large around the 1/4 point. If I try to speed up the perihelic angular velocity by increasing the eccentricity a bit compared to Kepler, the excess quadrature radius just gets worse. As I understand it, Kepler knew he could kludge these discrepancies out by piling on a series of higher-order small epicycles, but he concluded it was much better as an exercise in physics to find an analytically simple figure that would do the job, thus breaking once and for all with the ideal of uniform circular motion. After some hard mathematical slogging he found that the ellipse regulated by the equal area rule made a good fit with Tycho's positional data.

As I understand it, Kepler knew he could kludge these discrepancies out by piling on a series of higher-order small epicycles, but he concluded it was much better as an exercise in physics to find an analytically simple figure that would do the job, thus breaking once and for all with the ideal of uniform circular motion

"From the time of Newton, it has been known that Keplerís laws are mere approximations, computerís fictions, handy mathematical
devices for finding the approximate place of a planet in the heavens. They apply with greater accuracy to some planets than to
others. Jupiter and Saturn show the greatest deviations from strictly elliptical motion. The latter body is often nearly a degree
away from the place it would have been had its motion about the sun been strictly in accord with Keplerís laws. This is such a
large discrepancy that it can be detected by the unaided eye. The moon is approximately half a degree in diameter, so that the
discrepancy in the motion of Saturn is about twice the apparent diameter of the moon. In a single year, during the course of one
revolution about the sun, the Earth may depart from the theoretical ellipse by an amount sufficient to appreciably change
the apparent place of the sun in the heavens." (Charles Lane Poor, 'Gravitation versus Relativity' p129)

"From the time of Newton, it has been known that Kepler’s laws are mere approximations, computer’s fictions, handy mathematical
devices for finding the approximate place of a planet in the heavens. They apply with greater accuracy to some planets than to
others. Jupiter and Saturn show the greatest deviations from strictly elliptical motion. The latter body is often nearly a degree
away from the place it would have been had its motion about the sun been strictly in accord with Kepler’s laws. This is such a
large discrepancy that it can be detected by the unaided eye. The moon is approximately half a degree in diameter, so that the
discrepancy in the motion of Saturn is about twice the apparent diameter of the moon. In a single year, during the course of one
revolution about the sun, the Earth may depart from the theoretical ellipse by an amount sufficient to appreciably change
the apparent place of the sun in the heavens." (Charles Lane Poor, 'Gravitation versus Relativity' p129)

Did Mr. Poor say how much time was required for Saturn to depart by a degree from a best-fit Kepler ellipse? Tycho's data set covered roughly one orbit period of Saturn. Is Mr. Poor saying that Jupiter perturbed Saturn that severely during that time? If so, I think we would have seen mention of it in writings about Kepler.

The way I read that quote, I get the impression that the discrepancy
occurs about once each orbit, and is way, way too big to be just a
perturbation by the other planets. I have no clue what's going on.

The way I read that quote, I get the impression that the discrepancy
occurs about once each orbit, and is way, way too big to be just a
perturbation by the other planets. I have no clue what's going on.

-- Jeff, in Minneapolis

My hunch is an unreliable source. As I understand it, Kepler could get a good fit with Tycho's data with a cleaned up Ptolemaic or Copernican construction for Venus, Jupiter and Saturn. I don't know about Mercury, but Tycho would have had a hard time getting good data for most of its orbit. Mars was the rogue because of a double whammy of large eccentricity and close approaches to Earth. His equal area rule ellipse gave him good agreement with Tycho's data for Mars. It did not work for the Moon without adding on some large empirically determined periodic fudge terms, and we now know that severe perturbation by the Sun is the cause. Analogous fudge terms were not needed for the planets at arcminute precision because their mutual perturbations are proportionately vastly smaller.

Check out this animation.http://science.larouchepac.com/keple...opernicus.html
Some of the 34 circles he announced initially were superfluous by our construction standards. To construct his eccentric deferent he started with a radius vector revolving about the Sun at constant rate. The tip of this vector carried an epicycle vector that rotated retrograde with respect to the deferent vector, thus creating the eccentric circle with a stationary center in a non-rotating frame of reference. This epicycle carried a second smaller epicycle which rotated at twice the rate direct with respect to the other epicycle. Nowadays we reduce the clutter by simply constructing the eccentric circle in the stationary frame of reference.

So, my question is, does a Ptolemaic system without an equant require more circles than Copernicus's? If not, I can't really understand what Copernicus was trying to do.

"It is very important to acknowledge that the Copernican theory offers a very exact calculation of the apparent movements of the planets, even though it must be conceded that, from the modern standpoint practically identical results could be obtained by means of a somewhat revised Ptolemaic system. It makes no sense, accordingly, to speak of a difference in truth between Copernicus and Ptolemy: both conceptions are equally permissible descriptions. What has been considered as the greatest discovery of occidental wisdom, as opposed to that of antiquity, is questioned as to its truth value." ("From Copernicus to Einstein", Hans Reichenbach)